Add Wasserstein distance

Author: charleskawczynski

src/Histograms.jl

@@ -11,13 +11,10 @@ Functions in this module:
 """
 module Histograms
 
-using PyCall
 using NPZ
 using ..Utilities
 
-using Conda
-Conda.add("scipy")
-scsta = pyimport_conda("scipy.stats","")
+include("wasserstein.jl")
 
 """
 A simple struct to store samples for empirical PDFs (histograms, distances etc.)
@@ -196,13 +193,14 @@ Returns:
 """
 function W1(u_samples::AbstractVector, v_samples::AbstractVector;
                      normalize = false)
-  return if !normalize
-    scsta.wasserstein_distance(u_samples, v_samples)
+  d = wasserstein_distance(u_samples, v_samples)
+  if !normalize
+    return d
   else
     u_m, u_M = extrema(u_samples)
     v_m, v_M = extrema(v_samples)
     L = max(u_M, v_M) - min(u_m, v_m)
-    scsta.wasserstein_distance(u_samples, v_samples) / L
+    return d / L
   end
 end
 

src/wasserstein.jl

@@ -0,0 +1,77 @@
+#####
+##### Wasserstein distance
+#####
+
+function pysearchsorted(a,b;side="left")
+    if side == "left"
+        return searchsortedfirst.(Ref(a),b) .- 1
+    else
+        return searchsortedlast.(Ref(a),b)
+    end
+end
+
+function _cdf_distance(p, u_values, v_values, u_weights=nothing, v_weights=nothing)
+    _validate_distribution!(u_values, u_weights)
+    _validate_distribution!(v_values, v_weights)
+
+    u_sorter = sortperm(u_values)
+    v_sorter = sortperm(v_values)
+
+    all_values = vcat(u_values, v_values)
+    sort!(all_values)
+
+    # Compute the differences between pairs of successive values of u and v.
+    deltas = diff(all_values)
+
+    # Get the respective positions of the values of u and v among the values of
+    # both distributions.
+    u_cdf_indices = pysearchsorted(u_values[u_sorter],all_values[1:end-1], side="right")
+    v_cdf_indices = pysearchsorted(v_values[v_sorter],all_values[1:end-1], side="right")
+
+    # Calculate the CDFs of u and v using their weights, if specified.
+    if u_weights == nothing
+        u_cdf = (u_cdf_indices) / length(u_values)
+    else
+        u_sorted_cumweights = vcat([0], cumsum(u_weights[u_sorter]))
+        u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[end]
+    end
+
+    if v_weights == nothing
+        v_cdf = (v_cdf_indices) / length(v_values)
+    else
+        v_sorted_cumweights = vcat([0], cumsum(v_weights[v_sorter]))
+        v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[end]
+    end
+
+    # Compute the value of the integral based on the CDFs.
+    if p == 1
+        return sum(abs.(u_cdf - v_cdf) .* deltas)
+    end
+    if p == 2
+        return sqrt(sum((u_cdf - v_cdf).^2 .* deltas))
+    end
+    return sum(abs.(u_cdf - v_cdf).^p .* deltas)^(1/p)
+end
+
+function _validate_distribution!(vals, weights)
+    # Validate the value array.
+    length(vals) == 0 && throw(ValueError("Distribution can't be empty."))
+    # Validate the weight array, if specified.
+    if weights ≠ nothing
+        if length(weights) != length(vals)
+            throw(ValueError("Value and weight array-likes for the same
+                              empirical distribution must be of the same size."))
+        end
+        any(weights .< 0) && throw(ValueError("All weights must be non-negative."))
+        if !(0 < sum(weights) < Inf)
+            throw(ValueError("Weight array-like sum must be positive and
+                              finite. Set as None for an equal distribution of
+                              weight."))
+        end
+    end
+    return nothing
+end
+
+function wasserstein_distance(u_values, v_values, u_weights=nothing, v_weights=nothing)
+    return _cdf_distance(1, u_values, v_values, u_weights, v_weights)
+end